We have: #(- 4 + 2 i) / (6 - 2 i)#
First, let's multiply both the numerator and the denominator by the complex conjugate of the denominator:
#= (- 4 + 2 i) / (6 - 2 i) cdot (6 + 2 i) / (6 + 2 i)#
#= ((- 4) (6) + (- 4) (2 i) + (2 i) (6) + (2 i) (2 i)) / ((6)^(2) - (2 i)^(2))#
#= (- 24 - 8 i + 12 i + 4 i^(2)) / (36 - 4 i^(2))#
Let's apply the fact that #i^(2) = - 1#:
#= (- 24 + (4 cdot (- 1)) + 4 i) / (36 - (4 cdot (- 1)))#
#= (- 24 - 4 + 4 i) / (36 + 4)#
#= (- 28 + 4 i) / (40)#
#= - (7) / (10) + (1) / (10) i#
Then, we need to determine the modulus and the argument of this complex number.
Let #z = - (7) / (10) + (1) / (10) i#:
#=> |z| = sqrt((- (7) / (10))^(2) + ((1) / (10))^(2))#
#=> |z| = sqrt((49) / (100) + (1) / (100))#
#=> |z| = sqrt((50) / (100))#
#=> |z| = sqrt((1) / (2))#
#=> |z| = (sqrt(2)) / (2)#
and
#=> arg(z) = arctan(((1) / (10)) / (- (7) / (10)))#
#=> arg(z) = arctan(- (1) / (7))#
#=> arg(z) approx - 0.142#
Now, let's express the complex number in polar form:
#=> z = (sqrt(2)) / (2) (cos(- 0.412) + i sin(- 0.412))#
#=> z = (sqrt(2)) / (2) (cos(0.412) - i sin(0.412))#
